Simulating Index Event Bias

Simulation Parameters

Distribution And Effect Size Of Risk Factors

Data Generating Processes

The effect sizes of the risk factors are similar to those of the UMOD SNP.

f_0         <- function(t) 0.10 * t^2 / (0.7 + 0.04 * pmax(0, t - 3)^3)
g_0         <- function(t) 0.15 * t^2 / (0.9 + 0.01 * pmax(0, t - 1)^3)
f_1         <- function(t) 0.32 * exp(-0.15 * t)
f_until_1   <- function(t) 2.50 * exp(-0.60 * t)
g_1         <- function(t) 0.14 * exp(-0.25 * t)
g_until_1   <- function(t) 0.14 * exp(-0.25 * t)

f_0_1 <- f_0
f_0_3 <- g_0
f_1_2 <- function(t) 0.48 * exp(-0.10 * t)
f_1_3 <- function(t) 0.16 * exp(-0.30 * t)

formulas_dgp_timeScales <- list(
  list(from = 0, to = 1,
    formula = ~
      f_0(tend) + beta_0_01 + beta_1_01 * x1 + beta_2_01 * x2
  ),
  list(
    from = 0, to = 3,
    formula = ~
      g_0(tend) + beta_0_03 + beta_1_03 * x1 + beta_2_03 * x2
  ),
  list(
    from = 1, to = 2,
    formula = ~
      f_0(tend) + f_1(t_1) + f_until_1(t_until_1) + beta_0_12 + beta_1_12 * x1 + beta_2_12 * x2
  ),
  list(
    from = 1, to = 3,
    formula = ~
      g_0(tend) + g_1(t_1) + g_until_1(t_until_1) + beta_0_13 + beta_1_13 * x1 + beta_2_13 * x2
  )
)

formulas_dgp_stratified <- list(
  list(from = 0, to = 1,
    formula = ~
      f_0_1(tend) + beta_0_01 + beta_1_01 * x1 + beta_2_01 * x2
  ),
  list(
    from = 0, to = 3,
    formula = ~
      f_0_3(tend) + beta_0_01 + beta_1_03 * x1 + beta_2_03 * x2
  ),
  list(
    from = 1, to = 2,
    formula = ~
      f_1_2(tend) + f_until_1(t_until_1) + beta_0_01 + beta_1_12 * x1 + beta_2_12 * x2
  ),
  list(
    from = 1, to = 3,
    formula = ~
      f_1_3(tend) + g_until_1(t_until_1) + beta_0_01 + beta_1_13 * x1 + beta_2_13 * x2
  )
)

Other Parameters

cut <- seq(0, 10, by = 0.1)
terminal_states <- c(2, 3)
n <- 5000
round <- 1
cens_type <- "right"
cens_dist <- "weibull"
cens_params <- c(1.5, 10.0) # shape, scale
bs <- "ps"
k <- 20

Model Formulas

formula_mod_timeScales <- ped_status ~
  s(tend, by = trans_to_3, bs = bs, k = k) +
  s(t_1, by = trans_after_1, bs = bs, k = k) +
  s(t_until_1, by = trans_after_1, bs = bs, k = k) +
  transition * x1 + transition * x2

formula_mod_timeScales_ieb <- ped_status ~
  s(tend, by = trans_to_3, bs = bs, k = k) +
  s(t_1, by = trans_after_1, bs = bs, k = k) +
  s(t_until_1, by = trans_after_1, bs = bs, k = k) +
  transition * x1

formula_mod_stratified <- ped_status ~
  s(tend, by = transition, bs = bs, k = k) +
  s(t_until_1, by = trans_after_1, bs = bs, k = k) +
  transition * x1 + transition * x2

formula_mod_stratified_ieb <- ped_status ~
  s(tend, by = transition, bs = bs, k = k) +
  s(t_until_1, by = trans_after_1, bs = bs, k = k) +
  transition * x1

Correlations

Bernoulli-distributed omitted risk factor (p=0.5)

Bernoulli-distributed included risk factor (p=0.5)

Moderate Effect Sizes

Strong Effect Sizes

Normally distributed included risk factor (sd=1)

Moderate Effect Sizes

Strong Effect Sizes

Normally distributed included risk factor (sd=5)

Moderate Effect Sizes

Strong Effect Sizes

Normally distributed omitted risk factor (sd=1)

Bernoulli-distributed included risk factor (p=0.5)

Moderate Effect Sizes

Strong Effect Sizes

Normally distributed included risk factor (sd=1)

Moderate Effect Sizes

Strong Effect Sizes

Normally distributed included risk factor (sd=5)

Moderate Effect Sizes

Strong Effect Sizes

Normally distributed omitted risk factor (sd=5)

Bernoulli-distributed included risk factor (p=0.5)

Moderate Effect Sizes

Strong Effect Sizes

Normally distributed included risk factor (sd=1)

Moderate Effect Sizes

Strong Effect Sizes

Normally distributed included risk factor (sd=5)

Moderate Effect Sizes

Strong Effect Sizes

Conclusion

• No negative correlation in state 0 (by construction)
• No / little negative correlation in state 1 when omitted risk factor x2 is binary
• Negative correlation in state 1 when omitted risk factor x2 is normally distributed, increasing in size with increasing variance of x2, up to -0.40

Coefficients

Bernoulli-distributed omitted risk factor (p=0.5)

Bernoulli-distributed included risk factor (p=0.5)

Moderate Effect Sizes

Strong Effect Sizes

Normally distributed included risk factor (sd=1)

Moderate Effect Sizes

Strong Effect Sizes

Normally distributed included risk factor (sd=5)

Moderate Effect Sizes

Strong Effect Sizes

Normally distributed omitted risk factor (sd=1)

Bernoulli-distributed included risk factor (p=0.5)

Moderate Effect Sizes

Strong Effect Sizes

Normally distributed included risk factor (sd=1)

Moderate Effect Sizes

Strong Effect Sizes

Normally distributed included risk factor (sd=5)

Moderate Effect Sizes

Strong Effect Sizes

Normally distributed omitted risk factor (sd=5)

Bernoulli-distributed included risk factor (p=0.5)

Moderate Effect Sizes

Strong Effect Sizes

Normally distributed included risk factor (sd=1)

Moderate Effect Sizes

Strong Effect Sizes

Normally distributed included risk factor (sd=5)

Moderate Effect Sizes

Strong Effect Sizes

Conclusion

• No IEB when omitted risk factor x2 is binary
• IEB for both 0->1 and 1->2 when omitted risk factor x2 is normally distributed, increasing in size with increasing variance of x2, causing attenuation bias of up to 25% + 25% = 50%

Coverage

Bernoulli-distributed omitted risk factor (p=0.5)

Bernoulli-distributed included risk factor (p=0.5)

Moderate Effect Sizes

Strong Effect Sizes

Normally distributed included risk factor (sd=1)

Moderate Effect Sizes

Strong Effect Sizes

Normally distributed included risk factor (sd=5)

Moderate Effect Sizes

Strong Effect Sizes

Normally distributed omitted risk factor (sd=1)

Bernoulli-distributed included risk factor (p=0.5)

Moderate Effect Sizes

Strong Effect Sizes

Normally distributed included risk factor (sd=1)

Moderate Effect Sizes

Strong Effect Sizes

Normally distributed included risk factor (sd=5)

Moderate Effect Sizes

Strong Effect Sizes

Normally distributed omitted risk factor (sd=5)

Bernoulli-distributed included risk factor (p=0.5)

Moderate Effect Sizes

Strong Effect Sizes

Normally distributed included risk factor (sd=1)

Moderate Effect Sizes

Strong Effect Sizes

Normally distributed included risk factor (sd=5)

Moderate Effect Sizes

Strong Effect Sizes

Conclusion

• Good coverage when omitted risk factor x2 is binary
• Coverage worsens significantly when omitted risk factor x2 is normally distributed, even though the bias (see coefficient plots) is sometimes very small